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J. Opt. Soc. Am. B / Vol. 16, No. 11 / November 1999
Hermier et al.
Spatial quantum noise of semiconductor lasers
J.-P. Hermier, A. Bramati, A. Z. Khoury, and E. Giacobino
Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Ecole Normale Supérieure,
Centre National de la Recherche Scientifique, 4 place Jussieu, F-75252 Paris Cedex 05, France
J.-Ph. Poizat, T. J. Chang, and Ph. Grangier
Laboratoire Charles Fabry de l’Institut d’Optique, B.P. 147, F91403 Orsay Cedex, France
Received April 2, 1999; revised manuscript received May 26, 1999
The transverse distribution of intensity noise in the far field of semiconductor lasers has been experimentally
studied. For a single-mode edge-emitting laser, it has been found that a large amount of noise is present in
higher-order nonlasing transverse modes parallel to the diode junction. In the case of a spatially multimode
vertical-cavity surface-emitting laser, each mode exhibits a large noise, but these noises show strong anticorrelations. © 1999 Optical Society of America [S0740-3224(99)00909-1]
OCIS codes: 140.5960, 140.2020, 250.7260, 270.6570, 230.5440.
1. INTRODUCTION
An understanding of the noise of semiconductor lasers
has great importance in fields spanning from optical telecommunications to high-sensitivity spectroscopy. Intensity noise reduction below the shot-noise level (SNL) was
demonstrated first by Machida et al.1 and has been demonstrated since then by many others.2–4 In principle, the
pumping mechanism of semiconductor lasers allows a
faithful conversion of the sub-Poissonian electron statistics of the driving current to the photon statistics of the
outgoing light.5,6 This simple picture does not, however,
work for all laser diodes, and many lasers exhibit larger
noise. Possible reasons for this nonideal behavior are
currently under investigation. For example, the importance of multimode effects7 on noise behavior has been
demonstrated in the classical regime8–10 and recently in
the quantum regime for longitudinal2–4,11,12 and polarization13 modes. In these experiments various modes
were lasing, and the main effect was an anticorrelation
among them because of mode competition. Such anticorrelations have also been observed between adjacent
transverse modes with orthogonal polarizations in
vertical-cavity surface-emitting lasers (VCSEL’s).14–17
In this study we investigate the influence of transverse
modes on the intensity noise of semiconductor lasers.
Understanding the dynamics of higher-order transverse
modes is of great importance when optimizing the coupling of a laser diode into an optical fiber or when diode
pumping a solid-state laser.18 Moreover, investigating
noise properties of a laser is a very sensitive way to obtain
information about its dynamics. The appearance of noise
in a mode can often be interpreted as a precursor for mode
switching phenomena.
We have studied two different types of laser, namely,
an edge-emitting laser,19 and a VCSEL.17 We have analyzed the spatial distribution of noise within the transverse profile of the laser beam by measuring its intensity
noise after screening part of it in the far field with a mov0740-3224/99/112140-07$15.00
able razorblade (see Fig. 1). The fitting of the noise profile allows us to precisely identify the noisy spatial modes
and to infer the presence of correlations between transverse modes. The present paper develops and compares
the results originally presented in Refs. 17 and 19.
It will be shown below that the spatial noise behavior of
edge-emitting lasers and of VCSEL’s differs. For the VCSEL, the different transverse modes involved are above
threshold and have no mutual coherence, owing to their
different frequencies and polarizations. In contrast, the
results presented for an edge-emitting laser describe a
situation in which only the main mode is lasing and in
which the transverse noise profile is due to an interference effect between this lasing mode and subthreshold
higher-order transverse modes.
In Section 2 we introduce a simple model for describing
the spatial dependence of quantum noise for both lasers
under consideration. Then we present the experimental
results obtained for the VCSEL (Section 3) and for the
stripe laser (Section 4). Finally, the results obtained for
the noise dependencies with the driving current in both
lasers are presented and compared in Section 5.
2. MODEL
In this section we present the model describing the transverse distribution of quantum noise within the laser beam
in the two cases corresponding to the VCSEL and to the
edge-emitting laser diode. It is sufficient to consider only
the lowest two transverse modes (labeled 0 and 1),20 with
amplitudes given by
u 0 共 x 兲 ⫽ 共 2 /␲ 兲 1/4冑1 /w exp共 ⫺x 2 /w 2 兲 ,
(1)
u 1 共 x 兲 ⫽ 共 2 /␲ 兲 1/4冑1 /w2 共 x/w 兲 exp共 ⫺x 2 /w 2 兲 ,
(2)
where w is the size of the beam. The principle of the
present experiments is to cut a part of the beam by means
of a movable razorblade. Denoting by X the position of
© 1999 Optical Society of America
Hermier et al.
Vol. 16, No. 11 / November 1999 / J. Opt. Soc. Am. B
2141
具 ␦ n 2共 X 兲 典 ⫽ 具 n 2共 X 兲 典 ⫺ 具 n 共 X 兲 典 2 ⫽ T 0共 X 兲 具 n 0典 ⫹ T 1共 X 兲
⫻ 具 n 1典 ⫹ T 02共 X 兲 具 : ␦ n 02: 典 ⫹ T 12共 X 兲 具 : ␦ n 12: 典
⫹ 2T 0 共 X 兲 T 1 共 X 兲 具 ␦ n 0 ␦ n 1 典 ,
(5)
where : : means that the operators are normally ordered.
Let us define C, the degree of correlation between the two
modes:
Fig. 1. Experimental setup for the measurement of the spatial
intensity noise distribution.
C⫽
the razorblade with respect to the beam center, we can
conveniently define the quantities
T 0共 X 兲 ⫽
冕
⬁
u 0 2 共 x 兲 dx,
T 1共 X 兲 ⫽
X
Q 01共 X 兲 ⫽
冕
⬁
冕
⬁
u 1 2 共 x 兲 dx,
具 ␦ n 0␦ n 1典
共 具 ␦ n 0 2 典具 ␦ n 1 2 典 兲 1/2
.
(6)
In the case of perfect correlations, C is equal to 1,
whereas, in the case of perfect anticorrelations, C is equal
to ⫺1. Using this notation, we find that
具 ␦ n 2共 X 兲 典 ⫽ T 0共 X 兲 具 n 0典 ⫹ T 1共 X 兲 具 n 1典 ⫹ T 02共 X 兲 具 : ␦ n 02: 典
X
⫹ T 1 2 共 X 兲 具 : ␦ n 1 2 : 典 ⫹ 2T 0 共 X 兲 T 1 共 X 兲
u 0 共 x 兲 u 1 共 x 兲 dx.
⫻ C 共 具 ␦ n 0 2 典具 ␦ n 1 2 典 兲 1/2.
(3)
X
It is clear that T 0 (X) and T 1 (X) correspond to the fraction of the intensity of each mode after the blade, while
Q 01(X) is the truncated overlap between the two modes.
Owing to the normalization and the orthogonality of the
mode functions, one obviously has T 0 (⫺⬁) ⫽ T 1 (⫺⬁)
⫽ 1 and Q 01(⫺⬁) ⫽ 0.
To calculate the quantum noise of the detected beam,
one can choose from among various possible orderings of
the field operators. Symmetrical ordering is often preferred to deal with squeezing effects; however, this requires that one carefully take into account all the vacuum
modes that enter as a result of optical losses in the system. When chopping off the beam, this approach is not
very convenient, and it is much simpler to use normal ordering. In the latter case the normally ordered variances
are obtained from a calculation that is quite similar to the
classical one. Then one obtains the true quantum noise
by adding the shot noise (proportional to the mean intensity) to the result obtained through the normal ordering.21
As will be shown again below, squeezing effects are then
recovered as negative values of the normally ordered variances. This approach is very convenient in the present
context, and it will be used systematically below.
A. Vertical-Cavity Surface-Emitting Laser
We now consider the situation in which the two transverse modes are lasing. In the laser that is used, these
two modes are orthogonally polarized, and their frequencies are separated by ⬃100 GHz. We will thus incoherently add their intensities and neglect any beatnote between the two modes. From the definitions given above,
the photon number 具 n(X) 典 in the chopped beam can be
obtained simply as a function of the photon numbers 具 n 0 典
and 具 n 1 典 in the TEM00 and TEM01 modes when the blade
edge is at position X:
具 n 共 X 兲 典 ⫽ T 0共 X 兲 具 n 0典 ⫹ T 1共 X 兲 具 n 1典 .
(4)
Using standard methods,21 we can obtain the fluctuations
in the beam as
(7)
We define the normalized variances as v i
⫽ 具 : ␦ n i 2 : 典 / 具 n i 典 . The value of v i represents excess noise
if positive and squeezing if negative. The ratio between
the intensities of the two modes q is defined as q
⫽ 具 n 0 典 / 具 n 1 典 . All the parameters used in this model
have a physical significance and can be measured independently. The total intensity noise S(X) when the razorblade is at position X and normalized to the shot noise
of the full beam is thus
S共 X 兲 ⫽
1
兵 qT 0 共 X 兲 ⫹ T 1 共 X 兲
q⫹1
⫹ T 0 2 共 X 兲 qv 0 ⫹ T 1 2 共 X 兲 v 1
⫹ 2T 0 共 X 兲 T 1 共 X 兲 C 关 q 共 1 ⫹ v 0 兲共 1 ⫹ v 1 兲兴 1/2其 .
(8)
We note that S(X) is normalized, so the SNL at position X
varies with the beam intensity and is therefore given by
S SNL共 X 兲 ⫽
qT 0 共 X 兲 ⫹ T 1 共 X 兲
.
q⫹1
(9)
B. Edge-Emitting Laser
For this laser, the experimental situation described in
this subsection deals with a single lasing mode (mode 0).
Mode 1 is below threshold and therefore has a broad spectrum with a width that is larger than the free spectral
range of the laser cavity. Although the central frequencies of modes 0 and 1 are different, there is a frequency
overlap that allows interference between them. In a linearized approximation, the positive part E ( ⫹) (x) of the
electric-field operator at position x will be given by
E 共 ⫹兲 共 x 兲 ⫽ E 0 u 0 共 x 兲 ⫹ ␦ E 共0⫹兲 u 0 共 x 兲 ⫹ ␦ E 共1⫹兲 u 1 共 x 兲 ,
(10)
where E 0 is the mean field of mode 0, taken to be real,
( ⫹)
and ␦ E 0,1
are the fluctuation operators of modes 0 and 1.
Owing to the spectrum overlap, the amplitude of mode 1
has to be coherently added to the main field amplitude.
To first order, the intensity operator is thus
n 共 X 兲 ⫽ T 0 共 X 兲 E 0 2 ⫹ E 0 关 T 0 共 X 兲 ␦ P 0 ⫹ Q 01共 X 兲 ␦ P 1 兴 ,
(11)
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J. Opt. Soc. Am. B / Vol. 16, No. 11 / November 1999
( ⫹)
( ⫺)
where ␦ P 0,1 ⫽ ␦ E 0,1
⫹ ␦ E 0,1
are quadrature amplitude
operators in phase with the mean field E 0 . The quantum noise when the razorblade is at position X is then
given by
具 ␦ n 2 共 X 兲 典 ⫽ T 0 共 X 兲 E 0 2 ⫹ E 0 2 关 T 0 2 共 X 兲 具 : ␦ P 0 2 : 典 ⫹ Q 012 共 X 兲
⫻ 具 : ␦ P 1 2 : 典 ⫹ 2T 0 共 X 兲 Q 01共 X 兲 具 ␦ P 0 ␦ P 1 典 兴
⫽ T 0 共 X 兲 具 n 0 典 ⫹ T 0 2 共 X 兲 具 : ␦ n 0 2 : 典 ⫹ E 0 2 关 Q 012 共 X 兲
⫻ 具 : ␦ P 1 2 : 典 ⫹ 2T 0 共 X 兲 Q 01共 X 兲 具 ␦ P 0 ␦ P 1 典 兴 ,
(12)
where the meaning of the various terms is as for Eq. (5).
The extra terms involving the overlap function Q 01(X) are
clearly due to the fact that the main lasing mode acts here
as a local oscillator that homodynes the noise of the
higher-order modes.22
In this case the normalized variances are defined by
v i ⫽ 具 : ␦ P i 2 : 典 , and the normalized correlation is given by
C⫽
具 ␦ P 0␦ P 1典
关共 1 ⫹ v 0 兲共 1 ⫹ v 1 兲兴 1/2
.
(13)
3. EXPERIMENT WITH VERTICAL-CAVITY
SURFACE-EMITTING LASERS
A. Preliminaries
The VCSEL’s used in these experiments are highquantum-efficiency oxide-confined GaAs/AlGaAs VCSEL’s with a diameter of 7 ␮m. They are made at the
Department of Optoelectronics of the University of Ulm,
Ulm, Germany.23 They consist of carbon-doped p-type
AlGaAs/AlGaAs and silicon-doped n-type AlAs/AlGaAs
Bragg reflectors with pairs of quarter-wavelength-thick
layers. The top (bottom) mirror has a reflectivity of
99.8% (99%). The mirrors surround the three active
8-nm-thick GaAs quantum wells, the cladding layers, and
the oxide aperture that provides both current and optical
confinement. The devices are attached to a copper plate
by use of silver paste and have an emission wavelength of
⬃840 nm. They have a very low threshold of I th0
⫽ 1.2 mA and a fairly large quantum differential efficiency of 48%. Above this driving current, the laser operates first in a single TEM00 mode up to the threshold of
the first transverse mode, I th1 . Both modes then lase together before a regime, I ⬎ I th2 , is achieved, where the
next transverse mode starts to oscillate. In Subsection
3.B the region of interest is mainly between I th1 and I th2 ,
when two and only two modes are lasing.
B. Experimental Setup
To take advantage of the principle of the pump noise suppression, a low-noise homemade power supply with an appropriate electronic filter provides the regulated electrical
current that drives the VCSEL’s. The device is also thermally stabilized with an active temperature stabilization.
Thanks to this stabilization, we were able to operate the
device at a fixed temperature with a drift as small as
0.01 °C per hour. The light beam is collimated by an
antireflection-coated microscope objective located at a dis-
Hermier et al.
tance of 2 mm from the laser output. The objective has a
large numerical aperture (N.A., 0.6) to avoid optical
losses, which would deteriorate the squeezing. The blade
used to screen the beam is coated to avoid optical feedback into the laser. It is also mounted on a micrometric
xy translation stage controlled by a motor. The motor enables us to gradually cut the beam over 50 s.
To measure the intensity noise and the corresponding
shot noise the standard scheme is to perform a balanced
detection, by means of a pair of high-quantum-efficiency
balanced photodiodes, on both sides of a beam splitter.
The sum of the two photocurrents is proportional to the
intensity noise, while the difference is proportional to the
corresponding shot noise.24 Owing to the lasing of two
modes with orthogonal linear polarizations, the shot noise
obtained by a balanced detection would not be reliable.
So only one photodiode is used (FND100; bandwidth, 10
kHz–30 MHz; quantum efficiency, 90%). The SNL is obtained by balanced detection of a laser diode beam that
has an intensity noise of 0.5 dB below the shot noise in
the frequency range of 1–30 MHz. We carefully checked
the linear dependence of the calibrated shot-noise signal
with the optical power incident on the photodiodes. The
shot noise obtained by this method was in agreement
within 0.1 dB with the noise obtained by a thermal light
generating the same dc current on the photodiode. The
photodiode is connected via a low-noise homemade amplifier (with a National Semiconductor CLC425) and an electronic amplifier (Nucletude 4-40-1A) to a spectrum analyzer (Tektronix 2753P). With this setup the electronic
noise was more than 6 dB below the signal that we measured for a typical detected power of 1.5 mW. In our experiment we could also perform a spectral analysis of the
laser beam with a high-resolution monochromator (0.03
nm at 840-nm output). At the output of the monochromator a Glan polarizer (extinction ratio, 10⫺4 ) allows us
to measure the polarization of the modes.
C. Experimental Results
The experimental procedure is as follows. To compare
our experimental results with the theoretical predictions
of the model presented in Section 2, we first need to measure the size of the beam [see Eq. (1)]. It is measured
when the laser is single mode, that is, for a driving current I op that is at an intermediate level between the
threshold I th0 of the TEM00 mode and the threshold I th1 of
the TEM01 mode. Note that the noise reduction levels
given throughout this paper are corrected for the transmission of the optical components and the detectors’
quantum efficiency. However, in the case of the measurement on VCSEL’s, the losses of the microscope objective are not corrected for.
We then adjust the electrical driving current to have
the VCSEL operating with only two transverse modes,
i.e., I th1 ⬍ I op ⬍ I th2 . The two transverse modes are
TEM00 and TEM01 (or TEM10). Using a monochromator
and a polarizer, we can verify that TEM00 (mode 0) and
TEM01 (mode 1) have linear polarizations orthogonal to
each other. We also measure the intensity of each mode
(separating them with a Glan polarizer), which gives the
values of 具 n 0 典 and 具 n 1 典 . From the measured intensity
noise of each mode we determine the values of the excess
Hermier et al.
noises v 0 and v 1 . The correlation [Eq. (6)] can be determined by use of v 0 , v 1 , and the value of the total intensity noise 具 ␦ n 2 (⫺⬁) 典 .
Our model then provides the value of 具 ␦ n 2 (X) 典 for every position of the blade. Finally, we measure the total
intensity noise and the total intensity of the beam versus
the position of the blade and compare it with the prediction of the model. Note that the theoretical trace is not
obtained by an optimization procedure, since all the parameters are measured beforehand separately.
In Fig. 2(a) we have plotted the experimental results
and the predictions of our model for the noise at 15 MHz
of a first VCSEL. The anticorrelations are equal to
⫺0.75 for these curves. As expected, since v 0 and v 1 are
large, we observe large variations of the intensity noise
with the position of the blade. The agreement between
theory and experiment is also very good. In Fig. 2(b) we
have plotted the experimental results and the predictions
of our model for a second VCSEL, which presented a
higher anticorrelation parameter: C ⫽ ⫺0.98. The
variations of the intensity noise are even larger. If we
compare the results of Figs. 2(a) and 2(b), we also notice
that the shape of the curves depends on the value of the
correlations. The agreement between experimental re-
Fig. 2. Normalized intensity noise of a VCSEL versus the position of the razorblade, normalized to the beam size. The smooth
(solid) trace is the theoretical prediction, and the noisy trace corresponds to the experimental results. Trace (S) is the SNL (proportional to the intensity). The dashed curve shows the theoretical intensity noise corresponding to an overall beam attenuation
equal to the transmission of the razorblade. The fitting parameters are as follows: (a) For VCSEL 1: C ⫽ ⫺0.75, v 0
⫽ 25.82, v 1 ⫽ 330.9, and q ⫽ 6.13; (b) for VCSEL 2: C
⫽ ⫺0.98, v 0 ⫽ 80.2, v 1 ⫽ 3648, q ⫽ 34.8.
Vol. 16, No. 11 / November 1999 / J. Opt. Soc. Am. B
2143
sults and theoretical predictions is again very good.
These results confirm the validity of our hypothesis, particularly the fact that we have taken only two modes into
account and that we have neglected the contribution of
the nonlasing modes.
Note that the partial screening by the blade can reduce
the noise by more than a simple attenuation effect. This
is shown in Fig. 2, where for some ranges of the blade position the intensity noise goes clearly below the noise of
the full beam attenuated down to the actual intensity of
the cut beam. For various positions of the blade, the intensity noise can be considerably increased or decreased,
depending on the contributions of the two modes and
their correlation. Note also that, when the blade is exactly halfway through (X ⫽ 0), the intensity noise is
equal to the intensity noise of the beam uniformly attenuated by 50%. This result derives from the fact that intensity noise profiles of the different transverse modes are
even functions.
4. EXPERIMENT WITH EDGE-EMITTING
LASERS
A. Experimental Setup
The single-mode semiconductor laser that was used is a
Fabry–Perot, multiple-quantum-well, index-guided AlGaAs device (SDL 5411-G1) emitting at 810 nm. It has a
single lasing transverse mode. It is collimated by a highN.A. (0.65) aspherical lens and is stabilized by the 24%
feedback of an external grating. The zeroth order of the
grating couples out 60% of the light power. The external
cavity ensures truly single-longitudinal-mode operation.4
The threshold of the laser system is I th ⫽ 17 mA. The
differential quantum efficiency measured at the output of
the grating is 32%. In this configuration its intensity
noise is 16% below the SNL, and its longitudinal sidemode rejection rate is ⬎35 dB.4,25 The values of the
noises given below correspond to the real noise at the output of the grating.
The experimental setup19 is similar to the one used for
the VCSEL experiment (see Fig. 1). The intensity noise
detection is a standard balanced scheme that allows a
convenient comparison between the intensity noise and
the SNL.24 The noise analysis frequency of the results
presented below is 14 MHz, but the frequency dependence
of the effects is essentially flat within the bandwidth of
our detectors (2–20 MHz). Note that an edge-emitting
laser does not present a cylindrical symmetry. One obtains all the results presented below by scanning the razorblade along the direction parallel to the junction plane.
Owing to a different optical guiding, no interesting transverse effects take place in the direction perpendicular to
the diode junction.
B. Experimental Results
The results for a laser-diode driving current of I
⫽ 100 mA are displayed in Fig. 3. The intensity profile
presents the Gaussian shape of the TE00 mode. The SNL
that is proportional to the intensity therefore exhibits an
error function shape as the blade cuts the beam. As can
be seen in Fig. 3, the intensity noise has a very different
shape. This difference signifies that the noise transverse
distribution is different from the mean intensity distribu-
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J. Opt. Soc. Am. B / Vol. 16, No. 11 / November 1999
Fig. 3. Normalized intensity noise of an edge-emitting laser versus the position of the razorblade, normalized to the beam size.
Trace (S) is the SNL (proportional to the intensity). Trace (L)
[(R)] corresponds to the razorblade entering the beam from the
left (right). For each trace the solid curve is the best fit. Trace
(R) has been flipped over for convenient comparison with trace
(L). The fitting parameters are v 0 ⫽ ⫺0.10 ⫾ 0.02, v 1 ⫽ 14.9
⫾ 0.2, v 2 ⫽ 1 ⫾ 0.5, C ⫽ 0.23 ⫾ 0.02. The inset, a zoom of the
SNL-normalized noise of trace (L), shows that the maximum
squeezing is obtained when part of the beam is screened. For
each trace, the solid curve is the best fit.
tion and that noise is present in higher-order spatial
modes. Note that the noise maximum is around the X
⫽ 0 position of the blade. This corresponds to fluctuations in the beam position. To identify these modes, and
to quantify their noise and correlations, the experimental
scans were fitted with the simple theoretical model presented in Section 2.
When looking carefully at expression (12), which gives
the intensity noise as a function of the blade position, one
can observe that the correlation term changes sign, depending on whether the razorblade enters the beam from
the right or from the left, owing to the following equality:
Q 01共 X 兲 ⬅
冕
⫹⬁
X
u 0 共 x 兲 u 1 共 x 兲 dx ⫽ ⫺
冕
⫺X
⫺⬁
Hermier et al.
(R) 兵i.e., 关 (L) ⫹ (R) 兴 /2其 is first fitted with the correlation
coefficient set to zero. One obtains v 0 ⫽ ⫺0.10, v 1
⫽ 14.9, and v 2 ⫽ 4. One then fits each trace by using
the established values of these parameters to obtain the
value of the correlation coefficients. One obtains C L
⫽ ⫺0.23 ⫾ 0.02 and C R ⫽ 0.22 ⫾ 0.02 for traces (L) and
(R), respectively, where one should have C L ⫽ ⫺C R , according to Eq. (12). The negative value for v 0 indicates
10% squeezing.
Note that the fits include the noise power v 2 of the
second-order TE20 mode, although the contribution of this
mode was not given in the model, for clarity. All the correlations involving this TE20 mode have been set to zero.
However, the fits are not very sensitive to the value of v 2 ,
which explains that the uncertainty regarding its value is
quite large.
5. DEPENDENCY OF INTENSITY NOISES
WITH THE DRIVING CURRENT
In this section the behavior of the noise in the various
modes is examined as a function of the laser driving current.
In Fig. 4(a) are displayed the intensity noises of modes
0 and 1, and the total intensity noise of the VCSEL, for
values of the current for which only these two modes are
lasing. This corresponds to the situation discussed in
Section 3. We also measured the noises of modes 0 and
1, separating them with a Glan polarizer. As is qualitatively expected from an independent mode theory for
above-threshold modes,6 their noises decrease with the
u 0 共 x 兲 u 1 共 x 兲 dx.
(14)
The correlation term therefore has a different sign, depending on whether the noise is measured on the righthand or the left-hand half of the beam. This means that,
when the correlation is nonzero, the noise level on the
right-hand side of the beam is different from the noise
level on the left-hand side. This is indeed what was observed experimentally (see Fig. 3).
In Fig. 3 we compare a scan taken with the blade entering the beam from the right (R) and a scan taken with
the blade entering from the left (L). Their different
shapes show that the correlations are nonzero. The
maximum noise value is different for each scan, showing
that the intensity noises of the two halves are different.
This symmetry breaking is due either to a slight misalignment of the grating or to a small defect in the semiconductor laser. As for the VCSEL, these correlations
lead to the a priori surprising result that one can improve
the full beam squeezing by chopping off part of the beam,
as is shown in the inset of Fig. 3.
As can be seen from Fig. 3, the experimental results are
well fitted by the model. The fits are performed in the
following way. The arithmetic average of traces (L) and
Fig. 4. Normalized intensity noise of the total beam (points), of
mode 0 (squares), and of mode 1 (triangles) versus the driving
current (a) for the VCSEL and (b) for the edge-emitting laser.
Hermier et al.
driving current. The total intensity noise, which is measured by removal of the polarizer, is smaller than the
noise of any of the two modes, owing to their anticorrelations.
The noise levels v 0 and v 1 of modes 0 and 1, respectively, for the edge-emitting laser are plotted in Fig. 4(b)
as a function of the driving current. Here v 0 is measured
directly from the noise of the total beam, whereas v 1 is inferred from the fit. It can be seen that the noise v 1 increases with the driving current. For a subthreshold
mode, one can indeed expect the noise level to increase as
the driving current increases and to diverge at threshold.
According to the standard model for noise in laser diodes,6
this noise varies as 8g 2 /( g ⫺ ␣ ) 2 , where g is the laser
gain and ␣ the losses. Note that the noise v 0 of mode 0
goes below the shot noise but remains higher than the
theoretical value 1 ⫺ ␩ , where ␩ is the overall quantum
efficiency. The reason for this discrepancy is currently
under investigation and might be linked to transversemode effects.
The observed behavior of mode 1 as a function of driving current is therefore consistent with our claim that
this mode is above threshold in the VCSEL and below
threshold for the edge-emitting laser.
Vol. 16, No. 11 / November 1999 / J. Opt. Soc. Am. B
VCSEL samples. This research was supported by the
European Strategic Program for R & D in Information
Technology (Advanced Quantum Information Research
20029) and by the European Community Training Mobility Researchers Microlasers and Cavity Quantum Electrodynamics programs (contract ERBFMRX CT 96-00066).
A. Bramati’s work was supported by Training Mobility
Researchers fellowship ERBFMBI CT 950204.
J.-P. Hermier can be reached at the address on the title
page, by telephone at 33-1-4427-4393, or by e-mail at
hermier@spectro.jussieu.fr.
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6. CONCLUSIONS
We have analyzed in this paper the spatial distribution of
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can be very different from the transverse intensity distribution. We have found that these two types of semiconductor laser exhibit different behaviors. In a VCSEL the
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the intensity fluctuations of both modes. In a singlemode edge-emitting laser the effect is coherent, and the
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